So that (REF ) can be written as Taylor series. The special point above is \((\omega _1=-i2\pi T, k_1)\) . We extend pole-skipping phenomenon at higher Matsubara frequencies \(\omega _n=-i2\pi Tn\) , and the method has been showed in [1]}. Now we review this process calculating special points \((\omega _n, k_n)\) . Firstly we expand \(h^y_x(r)\)  with a Taylor series
\(h^y_x(r)=\sum ^\infty _{p=0}h^y_{xp}(r-r_+)^p=h^y_{x0}+h^y_{x1}(r-r_+)+h^y_{x2}(r-r_+)^2+\dots .\) 
